Screw Modifier with varying Offset
I don't know how to achieve this with the screw modifier, but this can be done with a modified helix curve and a circle as bevel object.
Method 1: Lattice
For this methods you will have to create a helix screw curve. This can be either done by creating a mesh with a single vertex, using the screw modifier and converting the mesh to a curve, or by using the included add-on Add Curve: Extra Objects which allows you to add a helix spiral by selecting Add &gt; Curve &gt; Profiles &gt; Helix (3D).
Now add a Lattice object and (in object mode) place it so that it surrounds the spring. In the Data tab of the Properties area set a sufficient number of subdivisions.

Add the Lattice modifier to the helix curve and make sure to turn on Apply on splines' points.

With the Lattice selected, go into edit mode and move the vertices to create the desired spacing.

Method 2: Calculate control points' position
Another approach would be to move the curve's control points to the final position, with either a script or with some add-on. The add-on Animation Nodes provides an interface to enter custom interpolation curves which makes it very convenient to change certain parameters of the spring. But of course it's also possible to use a more simple for loop instead.
The following Animation Nodes node setup will allow you to modify both the spring's spacing as well as the radius:


Link to .blend file with the node tree